Let $L$ be a line in $\mathbb{P}^2$ and $\Gamma_L$ be the subspace of $\mathbb{P}^5$ parametrizing the conics of $\mathbb{P}^2$ that are tangent to or contain $L$. Prove that $\Gamma_L$ is a quadric hypersurface of rank 3.

Let $(x_0,x_1,x_2)$ be the coordinates of $\mathbb{P}^2$ and $(z_0,z_1,z_2,z_3,z_4,z_5)$ of $\mathbb{P}^5$. Now, taking the line to be $x_0=0$ it is easy to see that the surface in $\mathbb{P}^5$ corresponding to the conics that contain the line is the plane $z_3=z_4=z_5=0$. But how does one prove that the hypersurface corresponding to conics tangent to the line is a quadric of rank 3? Further, what is its relation to this plane?

Taking coefficients of $1, \epsilon$ in $F(x_1,y_1)$ and $G(x_1,y_1)$ one gets four equations in $S=\{a,b,c,d,e,f,x_0,y_0,h_1,h_2\}$. Adding to these either $h_1 = 1$ or $h_2 = 1$ you have five equations from which you eliminate by groebner base calculation $T=\{x_0,y_0,h_1,h_2\}$. You get, going both ways, two polynomials $H_1(a,b,c,d,e,f) = H_2(a,b,c,d,e,f) = 0$.

These identical $H_1, H_2$ are the conditions on $a,b,c,d,e,f$ that $C$ is tangential to the line $X + Y - Z=0$.

It is (with Maple) a trivial computation to get the $6 \times 6$ matrix corresponding to $H_i$ and compute its rank. It is indeed $3$.

The above proof is not wholly rigorous though, as there is no justification that the tangency condition to $X + Y - Z = 0$ is generic (besides from reproducing the result sought for). An alternative would be to try the same computation with $p X + q Y - r Z = 0$ and indeterminates $p,q,r$.
I did this and got

It is of bidegree $(2,2)$ in $a,b,c,d,e,f$ and $p,q,r$.
The $6 \times 6$ matrix corresponding to $H_i$ now has quadratic polynomials in $p,q,r$ as entries. It is possible to compute its generic rank and one gets $3$ again.

Note, that the above calculation prescription with $x_1 = x_0 + h_1 \epsilon, y_1 = y_0 + h_2 \epsilon$ and the following groebner base calculation works for arbitrary $f(x_1,y_1) = 0 = g(x_1,y_1)$ and gives conditions of the existence of a common point $P$ with common tangent vector $v \in T_P g = T_P f$.